Finite Group Databasebeta

← #10817455

Subgroups — #10817455

Order 768.

926 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4795C211yesmaximal, characteristic
2C2×#4790C211yesmaximal, characteristic
3C2×C4×Dic12C211yesmaximal, characteristic
4#76492C211yesmaximal, characteristic
5#75981C211yesmaximal, characteristic
6C3×#1933C211yesmaximal, characteristic
7#70236C211yesmaximal, characteristic
8#70225C211yesmaximal, characteristic
9C8×Dic3⋊C4C211yesmaximal, characteristic
10#69690C211yesmaximal, characteristic
11#69631C211yesmaximal, characteristic
12#68908C211yesmaximal, characteristic
13#68851C211yesmaximal, characteristic
14#68795C211yesmaximal, characteristic
15#68781C211yesmaximal, characteristic
16#22998—31nomaximal, Sylow
17C2×Q8×Dic3C2211yescharacteristic
18C2×#944C2211yescharacteristic
19C22×#216C2211yescharacteristic
20C22×Dic12C2211yescharacteristic
21C2×#887C2211yescharacteristic
22C2×#887C2211yescharacteristic
23C2×C4×Dic6C2211yescharacteristic
24C2×C4×Dic6C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×C4×C8C2211yescharacteristic
28C3×#391C2211yescharacteristic
29C3×#389C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#818C2211yescharacteristic
32C2×#816C2211yescharacteristic
33C2×#816C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35C2×C8×Dic3C2211yescharacteristic
36C2×C8×Dic3C2211yescharacteristic
37#4984C2211yescharacteristic
38#4976—21no—
39#4975—21no—
40#4975C2211yescharacteristic
41Dic3×C4⋊C4C2211yescharacteristic
42#4973C2211yescharacteristic
43C2×#810C2211yescharacteristic
44C2×#810C2211yescharacteristic
45C2×#810C2211yescharacteristic
46C2×#807C2211yescharacteristic
47C4×Dic3⋊C4C2211yescharacteristic
48#4795C418yescomplemented
49#4790—24no—
50C4×Dic12—24no—
51#4655—21no—
52#4652—21no—
53#4649—21no—
54#4649—21no—
55#4647—21no—
56#4647—21no—
57C3×#340C2211yescharacteristic
58C3×#340C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62#4524C2211yescharacteristic
63C2×#462—31no—
64C2×C4×Q16—31no—
65C2×C4×Q16—31no—
66#2365—31no—
67Q8×C4⋊C4—31no—
68#1956—31no—
69#1947—31no—
70#1933—31no—
71#1933D311yescharacteristic
72#1878—31no—
73#1878—31no—
74#1863—31no—
75#1863—31no—
76C8×C4⋊C4—31no—
77C4×#103—31no—
78C22×C3×Q8C2311yescharacteristic
79C22×Dic6C2311yescharacteristic
80C22×Dic6C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C42C2311yescharacteristic
88Q8×Dic3—22no—
89Q8×Dic3—41no—
90#944—22no—
91#944C2×C414yes—
92C2×#216—22no—
93C2×#216—22no—
94C2×#216C2×C414yes—
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4D411yescharacteristic
108C2×Dic3⋊C4D411yescharacteristic
109C2×C4×Dic3—21no—
110C2×C4×Dic3—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3—21no—
114C2×C4×Dic3—21no—
115C2×C4×Dic3C2311yescharacteristic
116C2×C4×Dic3C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C2×Dic12—22no—
120C2×Dic12—22no—
121C2×Dic12C2×C414yes—
122#887—22no—
123#887—22no—
124#887—41no—
125#887C2×C414yes—
126C4×Dic6—22no—
127C4×Dic6—22no—
128C4×Dic6—41no—
129C4×Dic6—41no—
130C3×#103—22no—
131C3×#103C2×C414yes—
132C3×C4×C8—22no—
133C3×#95—21no—
134C3×#95—21no—
135C3×#95—21no—
136#835—41no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95D411yescharacteristic
142C3⋊#95D411yescharacteristic
143#818—22no—
144#816—22no—
145#816D414yes—
146Dic3⋊C8C2×C414yes—
147C8×Dic3—22no—
148C8×Dic3—22no—
149#810—22no—
150#810—41no—
151#810C2×C414yes—
152#807—22no—
153C22×Q16—31no—
154C22×Q16—31no—
155C2×C4⋊Q8—31no—
156C2×C4×Q8—31no—
157C2×C4×Q8—31no—
158C2×C4×Q8—31no—
159C2×C4×Q8—31no—
160C2×C4×Q8—31no—
161#462—38no—
162C4×Q16—64no—
163C4×Q16—64no—
164C2×#107—31no—
165C2×#107—31no—
166C2×C4⋊C8—31no—
167C2×#103—31no—
168C2×#103—31no—
169C2×#103—31no—
170C2×#103—31no—
171C2×#103—31no—
172C2×#103—31no—
173C2×#103C2×D311yescharacteristic
174C2×#103C2×D311yescharacteristic
175C2×C4×C8—31no—
176C2×C4×C8—31no—
177C2×C4×C8C2×D311yescharacteristic
178#391—31no—
179#391C2×D311yescharacteristic
180#389—31no—
181#389—61no—
182#389—61no—
183#389—61no—
184#389C2×D311yescharacteristic
185#387—61no—
186#387—61no—
187#384—31no—
188#384—61no—
189C4×C4⋊C4—31no—
190C4×C4⋊C4—31no—
191C4×C4⋊C4—61no—
192C4×C4⋊C4—61no—
193#340—31no—
194#340—31no—
195#340C2×D311yescharacteristic
196#340C2×D311yescharacteristic
197#336—31no—
198#336—31no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8C22×C412yes—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2411yescharacteristic
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3C2×D411yescharacteristic
217C22×Dic3C2×D411yescharacteristic
218C22×Dic3C2×D411yescharacteristic
219C22×Dic3C2×D411yescharacteristic
220C2×Dic6—21no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6C22×C412yes—
229C2×Dic6C22×C412yes—
230C2×Dic6C2×D412yes—
231C2×Dic6#4112yes—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8C22×C412yes—
237C2×C3×C8C2×D412yes—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—81no—
244C3×C4⋊C4C22×C412yes—
245C3×C42—21no—
246C3×C42—21no—
247#216—28no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—42no—
252C4⋊Dic3—81no—
253C4⋊Dic3C2×D412yes—
254Dic3⋊C4—24no—
255Dic3⋊C4—24no—
256Dic3⋊C4—24no—
257Dic3⋊C4—22no—
258Dic3⋊C4—42no—
259Dic3⋊C4—42no—
260Dic3⋊C4—42no—
261Dic3⋊C4—42no—
262Dic3⋊C4—81no—
263Dic3⋊C4Q1614yescomplemented
264C4×Dic3—21no—
265C4×Dic3—21no—
266C4×Dic3—24no—
267C4×Dic3—42no—
268C4×Dic3—42no—
269C4×Dic3—42no—
270C4×Dic3—42no—
271C4×Dic3—42no—
272C4×Dic3C22×C412yes—
273C4×Dic3#4112yes—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8C22×C412yes—
280Dic12—28no—
281C22×Q8—31no—
282C22×Q8—31no—
283C22×Q8C22×D311yescharacteristic
284C2×Q16—34no—
285C2×Q16—34no—
286C2×Q16—62no—
287C2×Q16—62no—
288C2×Q16—62no—
289C2×Q16—62no—
290C22×C8—31no—
291C22×C8—31no—
292C22×C8C22×D311yescharacteristic
293C22×C8C22×D311yescharacteristic
294C4⋊Q8—34no—
295C4⋊Q8—62no—
296C4×Q8—34no—
297C4×Q8—62no—
298C4×Q8—62no—
299C4×Q8—62no—
300C4×Q8—62no—
301C4×Q8—62no—
302C4×Q8—121no—
303C4×Q8—121no—
304C4×Q8—121no—
305C4×Q8—121no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—31no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4—61no—
317C2×C4⋊C4—61no—
318C2×C4⋊C4—61no—
319C2×C4⋊C4—61no—
320C2×C4⋊C4—61no—
321C2×C4⋊C4C22×D311yescharacteristic
322C2×C4⋊C4C22×D311yescharacteristic
323C2×C4⋊C4C22×D311yescharacteristic
324C2×C42—31no—
325C2×C42—31no—
326C2×C42—61no—
327C2×C42—61no—
328C2×C42—61no—
329C2×C42—61no—
330C2×C42—61no—
331C2×C42—61no—
332C2×C42C22×D311yescharacteristic
333#107—62no—
334#107—62no—
335C4⋊C8—34no—
336#103—22no—
337#103—34no—
338#103—34no—
339#103—62no—
340#103—62no—
341#103—121no—
342#103—121no—
343#103C4×D314yes—
344C4×C8—22no—
345C4×C8—62no—
346C4×C8—62no—
347#95—21no—
348#95—21no—
349#95—21no—
350#95—31no—
351#95—31no—
352#95—61no—
353#95—61no—
354#95—61no—
355#95—61no—
356C23×C3C22×D411yescharacteristic
357C3×Q8—22no—
358C3×Q8—22no—
359C3×Q8—41no—
360C3×Q8—41no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4C2×Q1612yes—
379C2×C3×C4C23×C411yescharacteristic
380C2×C3×C4C22×D411yescharacteristic
381C2×C3×C4C2×#4111yescharacteristic
382C2×C3×C4C2×#4111yescharacteristic
383C2×C3×C4H(5,𝔽2)11yescharacteristic
384C2×C3×C42−1+411yescharacteristic
385C2×Dic3—22no—
386C2×Dic3—21no—
387C2×Dic3—22no—
388C2×Dic3—22no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—22no—
393C2×Dic3—21no—
394C2×Dic3—22no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3C4×D412yes—
405C2×Dic3C4×D412yes—
406C2×Dic3C2×Q1612yes—
407C2×Dic3C2×Q1612yes—
408C2×Dic3#13512yes—
409Dic6—22no—
410Dic6—22no—
411Dic6—22no—
412Dic6—22no—
413Dic6—22no—
414Dic6—41no—
415Dic6—41no—
416Dic6C4×D414yes—
417C3×C8—22no—
418C3×C8—22no—
419C3⋊C8—22no—
420C3⋊C8—41no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—32no—
427C2×Q8—32no—
428C2×Q8—32no—
429C2×Q8—32no—
430C2×Q8—61no—
431C2×Q8—61no—
432C2×Q8—61no—
433C2×Q8—61no—
434C2×Q8—61no—
435C2×Q8—61no—
436C2×Q8—61no—
437C2×Q8—61no—
438C2×Q8C2×C4×D312yes—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—31no—
444C22×C4—31no—
445C22×C4—31no—
446C22×C4—31no—
447C22×C4—61no—
448C22×C4—61no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4D3×D411yescharacteristic
452C22×C4D3×D411yescharacteristic
453C22×C4C23×D311yescharacteristic
454Q16—68no—
455Q16—68no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—32no—
461C2×C8—61no—
462C2×C8—61no—
463C2×C8—61no—
464C2×C8—61no—
465C2×C8—61no—
466C2×C8C2×C4×D312yes—
467C2×C8D3×D412yes—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—34no—
474C4⋊C4—32no—
475C4⋊C4—62no—
476C4⋊C4—64no—
477C4⋊C4—64no—
478C4⋊C4—64no—
479C4⋊C4—61no—
480C4⋊C4—61no—
481C4⋊C4—61no—
482C4⋊C4—81no—
483C4⋊C4—122no—
484C4⋊C4—122no—
485C4⋊C4—122no—
486C4⋊C4—122no—
487C4⋊C4—122no—
488C4⋊C4—241no—
489C4⋊C4—241no—
490C4⋊C4C2×C4×D312yes—
491C42—21no—
492C42—21no—
493C42—32no—
494C42—32no—
495C42—64no—
496C42—61no—
497C42—61no—
498C42—122no—
499C42—122no—
500C42—122no—
501C42—122no—
502C42—122no—
503C22×C3C2×C4×D411yescharacteristic
504C22×C3D4211yescharacteristic
505C22×C3#54811yescharacteristic
506C22×C3C22×Q1611yescharacteristic
507C22×C3C2×#13511yescharacteristic
508C22×C3#57711yescharacteristic
509C22×C3#57811yescharacteristic
510C3×C4—22no—
511C3×C4—42no—
512C3×C4—42no—
513C3×C4—41no—
514C3×C4—81no—
515C3×C4—81no—
516C3×C4C2×C4×D411yescharacteristic
517C3×C4#52011yescharacteristic
518C3×C4#54711yescharacteristic
519C3×C4#55311yescharacteristic
520Dic3—24no—
521Dic3—22no—
522Dic3—22no—
523Dic3—42no—
524Dic3—41no—
525Dic3—42no—
526Dic3—81no—
527Dic3C4×Q1614yescomplemented
528C23C2×D3×D411yescharacteristic
529Q8—22no—
530Q8—22no—
531Q8—34no—
532Q8—41no—
533Q8—41no—
534Q8—62no—
535Q8—62no—
536Q8—62no—
537Q8—62no—
538Q8—62no—
539Q8—121no—
540Q8—121no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—32no—
549C2×C4—32no—
550C2×C4—32no—
551C2×C4—32no—
552C2×C4—32no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—62no—
564C2×C4—61no—
565C2×C4—61no—
566C2×C4—61no—
567C2×C4—62no—
568C2×C4—61no—
569C2×C4—62no—
570C2×C4—61no—
571C2×C4—61no—
572C2×C4—62no—
573C2×C4—61no—
574C2×C4—62no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4D3×Q1612yes—
583C2×C4C22×C4×D311yescharacteristic
584C2×C4C2×D3×D411yescharacteristic
585C2×C4D3×#4111yescharacteristic
586C2×C4D3×#4111yescharacteristic
587C2×C4#100911yescharacteristic
588C2×C4#101011yescharacteristic
589C8—22no—
590C8—22no—
591C8—62no—
592C8—121no—
593C2×C3#236111yescharacteristic
594C2×C3C2×C4×Q1611yescharacteristic
595C2×C3#294711yescharacteristic
596C2×C3#328411yescharacteristic
597C2×C3D4×Q1611yescharacteristic
598C2×C3#336311yescharacteristic
599C2×C3#336411yescharacteristic
600C22C4×D3×D411yescharacteristic
601C22#560911yescharacteristic
602C22#567711yescharacteristic
603C22#576211yescharacteristic
604C22C2×D3×Q1611yescharacteristic
605C22D3×#13511yescharacteristic
606C22#577111yescharacteristic
607C4—22no—
608C4—34no—
609C4—42no—
610C4—41no—
611C4—42no—
612C4—62no—
613C4—62no—
614C4—64no—
615C4—81no—
616C4—81no—
617C4—121no—
618C4—122no—
619C4—122no—
620C4—241no—
621C4#552311yescharacteristic
622C4C4×D3×D411yescharacteristic
623C4#565511yescharacteristic
624C4#570911yescharacteristic
625C3#2299811yescharacteristic, complemented, Sylow
626C2#7547711yescharacteristic
627C2#7964811yescharacteristic
628C2C4×D3×Q1611yescharacteristic
629C2#8067911yescharacteristic
630C2#8078011yescharacteristic
631C2#8162411yescharacteristic
632C2#8163411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1933C2
socle, prime radicalC23×C3C22×D4
cosocleC22×C3×C4C24
hypercenter#1933D3
hypocenterC3#22998
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1933

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
48. #4795C4semidirect
263. Dic3⋊C4Q16semidirect
527. Dic3C4×Q16semidirect
625. C3#22998semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#47952
C2×#47902
C2×C4×Dic122
#764922
#759812
C3×#19332
#702362
#702252
C8×Dic3⋊C42
#696902
#696312
#689082
#688512
#687952
#687812
#229983